It seems like a plot dump, but that's only because the manga is usually so stingy with them plot hints. It's actually perfectly comprehensible amount of exposition, especially since many things we've known or seen previously, even if only as little hints here and there.
Basically, Yukariko found 12 King's Jewels with her friends and they kept one each while giving the rest to Mikado, making him believe there were 9 stones total.
Yukariko was guided to the Royal Power which was then sealed in the coffin of the mansion and gave her stone to Himegami, who used it somehow.
The power then returned to the Royal Garden and many years later Himegami met with Mikado, informed him about the power and together they looked into ways to get to it. Mikado raised Maria (didn't expect her to be involved in any way) into a genius who'd be able to decipher ancient text and stuff, but in the end the path was successfully opened by Athena, her grief caused by the loss of her dad and inheritance issues.
But just as Mikado and Athena entered the Palace, King Midas attacked. Athena managed to get in, but Mikado didn't - and his seven stones (he'd given one to Hayate and one to Athena) lost their power as a result.
That's when the whole arc with A-tan and Hayate took place and we saw Hayate's jewel ended up in possession of his brother, who saved A-tan back then.
Because Mikado was certain the only active stones were Athena's and Hayate's, he was sure Hayate was the one who later stole the Royal Power - but it was Yukariko again, who used Saginomiya's (Isumi's mom's) stone. She used it somehow (going to the palace she said to Hayate's brother that "there was a friend there to be rescued") and it returned to the Palace yet again.
One of the stones was destroyed by Nagi in Mykonos.
The stone Hayate's brother had as well as the one Mikado himself had were recently destroyed by A-tan and Isumi duo working together. Also, one was destroyed when Mikado tried to open the path by himself. And Miado wanted that to happen, because his plans, whatever they are, could be foiled by grown-up Athena.
One of the stones Mikado had was purified by Isumi and ended up in Aika's hands. She wanted to get rid of it, so Mikado asked her to give it to Athena in Greece. She did exactly that, but a person who's already used the Royal Power cannot touch the light of the jewel again or they'll get cursed... And that's how Athena shrunk, her memories reverted to before she used the Power.
Long story short there are only five stones left - the one Mikoto (Wataru's mom) received, the one Athena hid somewhere and three stones Mikado gave out - one of them now held by Kananiwa (principal) and I assume the other two are in possession of... Well, I don't really recall, maybe we were told, but whatever. Point is, the principal intends to cause one of her students to open the path with grief, which is what the true purpose of the competition is.
Also there's the Hisui person who's after the Sanzenin inheritance and Mikado intends to do anything he can NOT to let her have it for some reason. Our antagonist.
Okay, it is quite a lot of info. But I can follow it okay and I suspect it's even simpler for people reading the manga in bunches instead of one chapter per week.